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Properties

Label 80.2.s.b.27.2
Level $80$
Weight $2$
Character 80.27
Analytic conductor $0.639$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,2,Mod(3,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 80.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.638803216170\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 27.2
Root \(0.0376504 - 1.41371i\) of defining polynomial
Character \(\chi\) \(=\) 80.27
Dual form 80.2.s.b.3.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.29924 - 0.558542i) q^{2} -2.55161 q^{3} +(1.37606 + 1.45136i) q^{4} +(1.49107 + 1.66635i) q^{5} +(3.31516 + 1.42518i) q^{6} +(2.40368 + 2.40368i) q^{7} +(-0.977191 - 2.65426i) q^{8} +3.51070 q^{9} +(-1.00653 - 2.99782i) q^{10} +(-2.67707 + 2.67707i) q^{11} +(-3.51117 - 3.70331i) q^{12} +2.40164i q^{13} +(-1.78040 - 4.46551i) q^{14} +(-3.80462 - 4.25187i) q^{15} +(-0.212908 + 3.99433i) q^{16} +(-0.0750544 - 0.0750544i) q^{17} +(-4.56125 - 1.96087i) q^{18} +(2.67236 - 2.67236i) q^{19} +(-0.366678 + 4.45708i) q^{20} +(-6.13324 - 6.13324i) q^{21} +(4.97342 - 1.98291i) q^{22} +(2.12375 - 2.12375i) q^{23} +(2.49341 + 6.77263i) q^{24} +(-0.553442 + 4.96928i) q^{25} +(1.34141 - 3.12031i) q^{26} -1.30310 q^{27} +(-0.180999 + 6.79621i) q^{28} +(-3.95795 - 3.95795i) q^{29} +(2.56827 + 7.64925i) q^{30} -1.65367i q^{31} +(2.50762 - 5.07068i) q^{32} +(6.83083 - 6.83083i) q^{33} +(0.0555929 + 0.139435i) q^{34} +(-0.421324 + 7.58941i) q^{35} +(4.83094 + 5.09530i) q^{36} -2.53082i q^{37} +(-4.96467 + 1.97942i) q^{38} -6.12803i q^{39} +(2.96587 - 5.58602i) q^{40} +1.70882i q^{41} +(4.54289 + 11.3942i) q^{42} +3.84601i q^{43} +(-7.56921 - 0.201586i) q^{44} +(5.23469 + 5.85005i) q^{45} +(-3.94547 + 1.57306i) q^{46} +(2.15264 - 2.15264i) q^{47} +(0.543256 - 10.1920i) q^{48} +4.55532i q^{49} +(3.49460 - 6.14717i) q^{50} +(0.191509 + 0.191509i) q^{51} +(-3.48565 + 3.30480i) q^{52} -1.29475 q^{53} +(1.69305 + 0.727839i) q^{54} +(-8.45262 - 0.469246i) q^{55} +(4.03113 - 8.72883i) q^{56} +(-6.81881 + 6.81881i) q^{57} +(2.93166 + 7.35302i) q^{58} +(5.29614 + 5.29614i) q^{59} +(0.935619 - 11.3727i) q^{60} +(10.2413 - 10.2413i) q^{61} +(-0.923645 + 2.14852i) q^{62} +(8.43858 + 8.43858i) q^{63} +(-6.09020 + 5.18744i) q^{64} +(-4.00197 + 3.58100i) q^{65} +(-12.6902 + 5.05960i) q^{66} -10.6230i q^{67} +(0.00565167 - 0.212211i) q^{68} +(-5.41898 + 5.41898i) q^{69} +(4.78640 - 9.62515i) q^{70} +2.27322 q^{71} +(-3.43062 - 9.31831i) q^{72} +(-9.99096 - 9.99096i) q^{73} +(-1.41357 + 3.28815i) q^{74} +(1.41217 - 12.6796i) q^{75} +(7.55589 + 0.201231i) q^{76} -12.8696 q^{77} +(-3.42276 + 7.96180i) q^{78} -8.70617 q^{79} +(-6.97341 + 5.60103i) q^{80} -7.20709 q^{81} +(0.954448 - 2.22017i) q^{82} +11.1310 q^{83} +(0.461838 - 17.3413i) q^{84} +(0.0131558 - 0.236978i) q^{85} +(2.14816 - 4.99689i) q^{86} +(10.0991 + 10.0991i) q^{87} +(9.72165 + 4.48963i) q^{88} +15.6390 q^{89} +(-3.53363 - 10.5244i) q^{90} +(-5.77276 + 5.77276i) q^{91} +(6.00475 + 0.159920i) q^{92} +4.21952i q^{93} +(-3.99914 + 1.59446i) q^{94} +(8.43775 + 0.468420i) q^{95} +(-6.39846 + 12.9384i) q^{96} +(5.00672 + 5.00672i) q^{97} +(2.54434 - 5.91846i) q^{98} +(-9.39839 + 9.39839i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{4} + 2 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{11} - 12 q^{14} - 20 q^{15} - 6 q^{17} - 24 q^{18} - 2 q^{19} - 12 q^{20} - 16 q^{21} + 12 q^{22} - 2 q^{23} - 4 q^{24} - 6 q^{25}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29924 − 0.558542i −0.918703 − 0.394949i
\(3\) −2.55161 −1.47317 −0.736586 − 0.676344i \(-0.763563\pi\)
−0.736586 + 0.676344i \(0.763563\pi\)
\(4\) 1.37606 + 1.45136i 0.688031 + 0.725681i
\(5\) 1.49107 + 1.66635i 0.666825 + 0.745214i
\(6\) 3.31516 + 1.42518i 1.35341 + 0.581827i
\(7\) 2.40368 + 2.40368i 0.908504 + 0.908504i 0.996152 − 0.0876474i \(-0.0279349\pi\)
−0.0876474 + 0.996152i \(0.527935\pi\)
\(8\) −0.977191 − 2.65426i −0.345489 − 0.938423i
\(9\) 3.51070 1.17023
\(10\) −1.00653 − 2.99782i −0.318293 − 0.947992i
\(11\) −2.67707 + 2.67707i −0.807167 + 0.807167i −0.984204 − 0.177037i \(-0.943349\pi\)
0.177037 + 0.984204i \(0.443349\pi\)
\(12\) −3.51117 − 3.70331i −1.01359 − 1.06905i
\(13\) 2.40164i 0.666094i 0.942910 + 0.333047i \(0.108077\pi\)
−0.942910 + 0.333047i \(0.891923\pi\)
\(14\) −1.78040 − 4.46551i −0.475833 − 1.19346i
\(15\) −3.80462 − 4.25187i −0.982348 − 1.09783i
\(16\) −0.212908 + 3.99433i −0.0532269 + 0.998582i
\(17\) −0.0750544 − 0.0750544i −0.0182034 − 0.0182034i 0.697947 − 0.716150i \(-0.254097\pi\)
−0.716150 + 0.697947i \(0.754097\pi\)
\(18\) −4.56125 − 1.96087i −1.07510 − 0.462182i
\(19\) 2.67236 − 2.67236i 0.613081 − 0.613081i −0.330666 − 0.943748i \(-0.607274\pi\)
0.943748 + 0.330666i \(0.107274\pi\)
\(20\) −0.366678 + 4.45708i −0.0819918 + 0.996633i
\(21\) −6.13324 − 6.13324i −1.33838 − 1.33838i
\(22\) 4.97342 − 1.98291i 1.06034 − 0.422757i
\(23\) 2.12375 − 2.12375i 0.442833 − 0.442833i −0.450130 − 0.892963i \(-0.648622\pi\)
0.892963 + 0.450130i \(0.148622\pi\)
\(24\) 2.49341 + 6.77263i 0.508965 + 1.38246i
\(25\) −0.553442 + 4.96928i −0.110688 + 0.993855i
\(26\) 1.34141 − 3.12031i 0.263073 − 0.611943i
\(27\) −1.30310 −0.250783
\(28\) −0.180999 + 6.79621i −0.0342056 + 1.28436i
\(29\) −3.95795 − 3.95795i −0.734974 − 0.734974i 0.236627 − 0.971601i \(-0.423958\pi\)
−0.971601 + 0.236627i \(0.923958\pi\)
\(30\) 2.56827 + 7.64925i 0.468900 + 1.39656i
\(31\) − 1.65367i − 0.297008i −0.988912 − 0.148504i \(-0.952554\pi\)
0.988912 − 0.148504i \(-0.0474458\pi\)
\(32\) 2.50762 − 5.07068i 0.443289 − 0.896379i
\(33\) 6.83083 − 6.83083i 1.18909 − 1.18909i
\(34\) 0.0555929 + 0.139435i 0.00953410 + 0.0239129i
\(35\) −0.421324 + 7.58941i −0.0712168 + 1.28284i
\(36\) 4.83094 + 5.09530i 0.805157 + 0.849216i
\(37\) − 2.53082i − 0.416064i −0.978122 − 0.208032i \(-0.933294\pi\)
0.978122 − 0.208032i \(-0.0667059\pi\)
\(38\) −4.96467 + 1.97942i −0.805375 + 0.321104i
\(39\) − 6.12803i − 0.981271i
\(40\) 2.96587 − 5.58602i 0.468945 − 0.883227i
\(41\) 1.70882i 0.266873i 0.991057 + 0.133436i \(0.0426012\pi\)
−0.991057 + 0.133436i \(0.957399\pi\)
\(42\) 4.54289 + 11.3942i 0.700983 + 1.75817i
\(43\) 3.84601i 0.586510i 0.956034 + 0.293255i \(0.0947386\pi\)
−0.956034 + 0.293255i \(0.905261\pi\)
\(44\) −7.56921 − 0.201586i −1.14110 − 0.0303902i
\(45\) 5.23469 + 5.85005i 0.780341 + 0.872074i
\(46\) −3.94547 + 1.57306i −0.581728 + 0.231936i
\(47\) 2.15264 − 2.15264i 0.313995 − 0.313995i −0.532460 − 0.846455i \(-0.678732\pi\)
0.846455 + 0.532460i \(0.178732\pi\)
\(48\) 0.543256 − 10.1920i 0.0784123 − 1.47108i
\(49\) 4.55532i 0.650760i
\(50\) 3.49460 − 6.14717i 0.494212 − 0.869342i
\(51\) 0.191509 + 0.191509i 0.0268167 + 0.0268167i
\(52\) −3.48565 + 3.30480i −0.483372 + 0.458293i
\(53\) −1.29475 −0.177848 −0.0889239 − 0.996038i \(-0.528343\pi\)
−0.0889239 + 0.996038i \(0.528343\pi\)
\(54\) 1.69305 + 0.727839i 0.230395 + 0.0990463i
\(55\) −8.45262 − 0.469246i −1.13975 − 0.0632731i
\(56\) 4.03113 − 8.72883i 0.538683 − 1.16644i
\(57\) −6.81881 + 6.81881i −0.903174 + 0.903174i
\(58\) 2.93166 + 7.35302i 0.384946 + 0.965499i
\(59\) 5.29614 + 5.29614i 0.689499 + 0.689499i 0.962121 − 0.272622i \(-0.0878908\pi\)
−0.272622 + 0.962121i \(0.587891\pi\)
\(60\) 0.935619 − 11.3727i 0.120788 − 1.46821i
\(61\) 10.2413 − 10.2413i 1.31126 − 1.31126i 0.390780 − 0.920484i \(-0.372205\pi\)
0.920484 − 0.390780i \(-0.127795\pi\)
\(62\) −0.923645 + 2.14852i −0.117303 + 0.272862i
\(63\) 8.43858 + 8.43858i 1.06316 + 1.06316i
\(64\) −6.09020 + 5.18744i −0.761274 + 0.648430i
\(65\) −4.00197 + 3.58100i −0.496383 + 0.444168i
\(66\) −12.6902 + 5.05960i −1.56206 + 0.622794i
\(67\) − 10.6230i − 1.29780i −0.760873 − 0.648901i \(-0.775229\pi\)
0.760873 − 0.648901i \(-0.224771\pi\)
\(68\) 0.00565167 − 0.212211i 0.000685365 − 0.0257343i
\(69\) −5.41898 + 5.41898i −0.652369 + 0.652369i
\(70\) 4.78640 − 9.62515i 0.572085 − 1.15043i
\(71\) 2.27322 0.269781 0.134891 − 0.990860i \(-0.456932\pi\)
0.134891 + 0.990860i \(0.456932\pi\)
\(72\) −3.43062 − 9.31831i −0.404303 − 1.09817i
\(73\) −9.99096 − 9.99096i −1.16935 − 1.16935i −0.982361 − 0.186992i \(-0.940126\pi\)
−0.186992 − 0.982361i \(-0.559874\pi\)
\(74\) −1.41357 + 3.28815i −0.164324 + 0.382240i
\(75\) 1.41217 − 12.6796i 0.163063 − 1.46412i
\(76\) 7.55589 + 0.201231i 0.866720 + 0.0230828i
\(77\) −12.8696 −1.46663
\(78\) −3.42276 + 7.96180i −0.387552 + 0.901496i
\(79\) −8.70617 −0.979520 −0.489760 − 0.871857i \(-0.662916\pi\)
−0.489760 + 0.871857i \(0.662916\pi\)
\(80\) −6.97341 + 5.60103i −0.779651 + 0.626214i
\(81\) −7.20709 −0.800787
\(82\) 0.954448 − 2.22017i 0.105401 − 0.245177i
\(83\) 11.1310 1.22178 0.610890 − 0.791715i \(-0.290812\pi\)
0.610890 + 0.791715i \(0.290812\pi\)
\(84\) 0.461838 − 17.3413i 0.0503907 − 1.89209i
\(85\) 0.0131558 − 0.236978i 0.00142695 − 0.0257039i
\(86\) 2.14816 − 4.99689i 0.231642 − 0.538829i
\(87\) 10.0991 + 10.0991i 1.08274 + 1.08274i
\(88\) 9.72165 + 4.48963i 1.03633 + 0.478596i
\(89\) 15.6390 1.65773 0.828866 − 0.559447i \(-0.188986\pi\)
0.828866 + 0.559447i \(0.188986\pi\)
\(90\) −3.53363 − 10.5244i −0.372477 − 1.10937i
\(91\) −5.77276 + 5.77276i −0.605149 + 0.605149i
\(92\) 6.00475 + 0.159920i 0.626038 + 0.0166729i
\(93\) 4.21952i 0.437544i
\(94\) −3.99914 + 1.59446i −0.412480 + 0.164456i
\(95\) 8.43775 + 0.468420i 0.865695 + 0.0480589i
\(96\) −6.39846 + 12.9384i −0.653040 + 1.32052i
\(97\) 5.00672 + 5.00672i 0.508355 + 0.508355i 0.914021 − 0.405666i \(-0.132960\pi\)
−0.405666 + 0.914021i \(0.632960\pi\)
\(98\) 2.54434 − 5.91846i 0.257017 − 0.597855i
\(99\) −9.39839 + 9.39839i −0.944573 + 0.944573i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 80.2.s.b.27.2 yes 18
3.2 odd 2 720.2.z.g.667.8 18
4.3 odd 2 320.2.s.b.207.8 18
5.2 odd 4 400.2.j.d.43.6 18
5.3 odd 4 80.2.j.b.43.4 ✓ 18
5.4 even 2 400.2.s.d.107.8 18
8.3 odd 2 640.2.s.c.287.2 18
8.5 even 2 640.2.s.d.287.8 18
15.8 even 4 720.2.bd.g.523.6 18
16.3 odd 4 80.2.j.b.67.4 yes 18
16.5 even 4 640.2.j.c.607.8 18
16.11 odd 4 640.2.j.d.607.2 18
16.13 even 4 320.2.j.b.47.2 18
20.3 even 4 320.2.j.b.143.8 18
20.7 even 4 1600.2.j.d.143.2 18
20.19 odd 2 1600.2.s.d.207.2 18
40.3 even 4 640.2.j.c.543.2 18
40.13 odd 4 640.2.j.d.543.8 18
48.35 even 4 720.2.bd.g.307.6 18
80.3 even 4 inner 80.2.s.b.3.2 yes 18
80.13 odd 4 320.2.s.b.303.8 18
80.19 odd 4 400.2.j.d.307.6 18
80.29 even 4 1600.2.j.d.1007.8 18
80.43 even 4 640.2.s.d.223.8 18
80.53 odd 4 640.2.s.c.223.2 18
80.67 even 4 400.2.s.d.243.8 18
80.77 odd 4 1600.2.s.d.943.2 18
240.83 odd 4 720.2.z.g.163.8 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.4 ✓ 18 5.3 odd 4
80.2.j.b.67.4 yes 18 16.3 odd 4
80.2.s.b.3.2 yes 18 80.3 even 4 inner
80.2.s.b.27.2 yes 18 1.1 even 1 trivial
320.2.j.b.47.2 18 16.13 even 4
320.2.j.b.143.8 18 20.3 even 4
320.2.s.b.207.8 18 4.3 odd 2
320.2.s.b.303.8 18 80.13 odd 4
400.2.j.d.43.6 18 5.2 odd 4
400.2.j.d.307.6 18 80.19 odd 4
400.2.s.d.107.8 18 5.4 even 2
400.2.s.d.243.8 18 80.67 even 4
640.2.j.c.543.2 18 40.3 even 4
640.2.j.c.607.8 18 16.5 even 4
640.2.j.d.543.8 18 40.13 odd 4
640.2.j.d.607.2 18 16.11 odd 4
640.2.s.c.223.2 18 80.53 odd 4
640.2.s.c.287.2 18 8.3 odd 2
640.2.s.d.223.8 18 80.43 even 4
640.2.s.d.287.8 18 8.5 even 2
720.2.z.g.163.8 18 240.83 odd 4
720.2.z.g.667.8 18 3.2 odd 2
720.2.bd.g.307.6 18 48.35 even 4
720.2.bd.g.523.6 18 15.8 even 4
1600.2.j.d.143.2 18 20.7 even 4
1600.2.j.d.1007.8 18 80.29 even 4
1600.2.s.d.207.2 18 20.19 odd 2
1600.2.s.d.943.2 18 80.77 odd 4